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NASA NTRS2008

Measuring Positions of Objects using Two or More Cameras

An improved method of computing positions of objects from digitized images acquired by two or more cameras (see figure) has been developed for use in tracking debris shed by a spacecraft during and shortly after launch. The method is also readily adaptable to such applications as (1) tracking moving and possibly interacting objects in other settings in order to determine causes of accidents and (2) measuring positions of stationary objects, as in surveying. Images acquired by cameras fixed to the ground and/or cameras mounted on tracking telescopes can be used in this method. In this method, processing of image data starts with creation of detailed computer- aided design (CAD) models of the objects to be tracked. By rotating, translating, resizing, and overlaying the models with digitized camera images, parameters that characterize the position and orientation of the camera can be determined. The final position error depends on how well the centroids of the objects in the images are measured; how accurately the centroids are interpolated for synchronization of cameras; and how effectively matches are made to determine rotation, scaling, and translation parameters. The method involves use of the perspective camera model (also denoted the point camera model), which is one of several mathematical models developed over the years to represent the relationships between external coordinates of objects and the coordinates of the objects as they appear on the image plane in a camera. The method also involves extensive use of the affine camera model, in which the distance from the camera to an object (or to a small feature on an object) is assumed to be much greater than the size of the object (or feature), resulting in a truly two-dimensional image. The affine camera model does not require advance knowledge of the positions and orientations of the cameras. This is because ultimately, positions and orientations of the cameras and of all objects are computed in a coordinate system attached to one object as defined in its CAD model.

Klinko, Steve↗
NASA NTRS2009

Apollo Onboard Navigation Techniques

This viewgraph presentation reviews basic navigation concepts, describes coordinate systems and identifies attitude determination techniques including Primary Guidance, Navigation and Control System (PGNCS) IMU management and Command and Service Module Stabilization and Control System/Lunar Module (LM) Abort Guidance System (AGS) attitude management. The presentation also identifies state vector determination techniques, including PGNCS coasting flight navigation, PGNCS powered flight navigation and LM AGS navigation.

Interbartolo, Michael↗
NASA NTRS2009

Estimating Relative Positions of Outer-Space Structures

A computer program estimates the relative position and orientation of two structures from measurements, made by use of electronic cameras and laser range finders on one structure, of distances and angular positions of fiducial objects on the other structure. The program was written specifically for use in determining errors in the alignment of large structures deployed in outer space from a space shuttle. The program is based partly on equations for transformations among the various coordinate systems involved in the measurements and on equations that account for errors in the transformation operators. It computes a least-squares estimate of the relative position and orientation. Sequential least-squares estimates, acquired at a measurement rate of 4 Hz, are averaged by passing them through a fourth-order Butterworth filter. The program is executed in a computer aboard the space shuttle, and its position and orientation estimates are displayed to astronauts on a graphical user interface.

Balian, Harry↗
NASA NTRS2010

Data Access Services that Make Remote Sensing Data Easier to Use

This slide presentation reviews some of the processes that NASA uses to make the remote sensing data easy to use over the World Wide Web. This work involves much research into data formats, geolocation structures and quality indicators, often to be followed by coding a preprocessing program. Only then are the data usable within the analysis tool of choice. The Goddard Earth Sciences Data and Information Services Center is deploying a variety of data access services that are designed to dramatically shorten the time consumed in the data preparation step. On-the-fly conversion to the standard network Common Data Form (netCDF) format with Climate-Forecast (CF) conventions imposes a standard coordinate system framework that makes data instantly readable through several tools, such as the Integrated Data Viewer, Gridded Analysis and Display System, Panoply and Ferret. A similar benefit is achieved by serving data through the Open Source Project for a Network Data Access Protocol (OPeNDAP), which also provides subsetting. The Data Quality Screening Service goes a step further in filtering out data points based on quality control flags, based on science team recommendations or user-specified criteria. Further still is the Giovanni online analysis system which goes beyond handling formatting and quality to provide visualization and basic statistics of the data. This general approach of automating the preparation steps has the important added benefit of enabling use of the data by non-human users (i.e., computer programs), which often make sub-optimal use of the available data due to the need to hard-code data preparation on the client side.

Lynnes, Christopher↗
NASA NTRS2011

Wind and Temperature Spectrometry of the Upper Atmosphere in Low-Earth Orbit

Wind and Temperature Spectrometry (WATS) is a new approach to measure the full wind vector, temperature, and relative densities of major neutral species in the Earth's thermosphere. The method uses an energy-angle spectrometer moving through the tenuous upper atmosphere to measure directly the angular and energy distributions of the air stream that enters the spectrometer. The angular distribution gives the direction of the total velocity of the air entering the spectrometer, and the energy distribution gives the magnitude of the total velocity. The wind velocity vector is uniquely determined since the measured total velocity depends on the wind vector and the orbiting velocity vector. The orbiting spectrometer moves supersonically, Mach 8 or greater, through the air and must point within a few degrees of its orbital velocity vector (the ram direction). Pointing knowledge is critical; for example, pointing errors 0.1 lead to errors of about 10 m/s in the wind. The WATS method may also be applied without modification to measure the ion-drift vector, ion temperature, and relative ion densities of major ionic species in the ionosphere. In such an application it may be called IDTS: Ion-Drift Temperature Spectrometry. A spectrometer-based coordinate system with one axis instantaneously pointing along the ram direction makes it possible to transform the Maxwellian velocity distribution of the air molecules to a Maxwellian energy-angle distribution for the molecular flux entering the spectrometer. This implementation of WATS is called the gas kinetic method (GKM) because it is applied to the case of the Maxwellian distribution. The WATS method follows from the recognition that in a supersonic platform moving at 8,000 m/s, the measurement of small wind velocities in the air on the order of a few 100 m/s and less requires precise knowledge of the angle of incidence of the neutral atoms and molecules. The same is true for the case of ion-drift measurements. WATS also provides a general approach that can obtain non-equilibrium distributions as may exist in the upper regions of the thermosphere, above 500 km and into the exosphere. Finally, WATS serves as a mass spectrometer, with very low mass resolution of roughly 1 part in 3, but easily separating atomic oxygen from molecular nitrogen.

Herrero, Federico↗
NASA NTRS2010

Selection Effects in Identifying Magnetic Clouds and the Importance of the Closest Approach Parameter

This study is motivated by the unusually low number of magnetic clouds (MCs) that are strictly identified within interplanetary coronal mass ejections (ICMEs), as observed at 1 AU; this is usually estimated to be around 30% or lower. But a looser definition of MCs may significantly increase this percentage. Another motivation is the unexpected shape of the occurrence distribution of the observers' "closest approach distances" (measured from a MC's axis, and called CA) which drops off somewhat rapidly as |CA| (in % of MC radius) approaches 100%, based on earlier studies. We suggest, for various geometrical and physical reasons, that the |CA|-distribution should be somewhere between a uniform one and the one actually observed, and therefore the 30% estimate should be higher. So we ask, When there is a failure to identify a MC within an ICME, is it occasionally due to a large |CA| passage, making MC identification more difficult, i.e., is it due to an event selection effect? In attempting to answer this question we examine WIND data to obtain an accurate distribution of the number of MCs vs. |CA| distance, whether the event is ICME-related or not, where initially a large number of cases (N=98) are considered. This gives a frequence distribution that is far from uniform, confirming earlier studies. This along with the fact that there are many ICME identification-parameters that do not depend on |CA| suggest that, indeed an MC event selection effect may explain at least part of the low ratio of (No. MCs)/(No. ICMEs). We also show that there is an acceptable geometrical and physical consistency in the relationships for both average "normalized" magnetic field intensity change and field direction change vs. |CA| within a MC, suggesting that our estimates of |CA|, B(sub 0) (magnetic field intensity on the axis), and choice of a proper "cloud coordinate" system (all needed in the analysis) are acceptably accurate. Therefore the MC fitting model (Lepping et al., 1990) is adequate, on average, for our analysis. However, this selection effect is not likely to completely answer our original question, on the unexpected ratio of MCs to ICMEs, so we must look for other factors, such as peculiarities of CME birth conditions. As a by-product of this analysis, we determine that the first order structural effects within a MC due to its interaction with the solar wind, plus the MC's usual expansion at 1 AU (i.e., the non-force free components of the MC's field) are, on average, weakly dependent on radial distance from the MC's axis; that is, in the outer reaches of a typical MC the non-force free effects show up, but even there they are rather weak. Finally, we show that it is not likely that a MC's size distribution statistically controls the occurrence distribution of the estimated |CA|s.

Lepping, R. P.↗
NASA NTRS2011

Toward Realistic Dynamics of Rotating Orbital Debris, and Implications for Lightcurve Interpretation

Optical observations of rotating space debris near GEO contain important information on size, shape, composition, and rotational states, but these aspects are difficult to extract due to data limitations and the high number of degrees of freedom in the modeling process. For tri-axial rigid debris objects created by satellite fragmentations, the most likely initial rotation state has a large component of initial angular velocity directed along the intermediate axis of inertia, leading to large angular reorientations of the body on the timescale of the rotation period. This lends some support to the simplest possible interpretation of light curves -- that they represent sets of random orientations of the objects of study. However, effects of internal friction and solar radiation are likely to cause significant modification of rotation states within a time as short as a few orbital periods. In order to examine the rotational dynamics of debris objects under the influences of these effects, a set of seven first-order coupled equations of motion were assembled in state form: three are Euler equations describing the rates of change of the components of angular velocity in the body frame, and four describe the rates of change of the components of the unit quaternion. Quaternions are a four-dimensional extension of complex numbers that form a seamless, singularity-free representation of body orientation on S3. The Euler equations contain explicit terms describing torque from solar radiation in terms of spherical harmonics, and terms representing effects of a prescribed rate of internal friction. Numerical integrations of these equations of motion are being performed, and results will be presented. Initial tests show that internal friction without solar radiation torque leads to rotation about the maximum principal axis of inertia, as required, and solar radiation torque is expected to lead to spin-up of objects. Because the axis of maximum rotational inertia tends to be roughly coincident with the normal to the largest projected cross-sectional area, internal friction is expected to lead to reduced variation of light curve amplitudes at a given phase angle, but a large dependence of the same on phase angle. At a given phase angle, databases are generated which contain reflected intensities for comprehensive sets of equally-likely orientations, represented as unit quaternions. When projected onto three dimensions (S2) and color-coded by intensity, the set is depicted as points within a solid, semi-transparent unit sphere, within which all possible reflected intensities for an object at a given phase angle may be inspected simultaneously. Rotational sequences are represented by trajectories through the sphere. Databases are generated for each of a set of phase angles separately, forming a comprehensive dataset of reflected intensities spanning all object orientations and solar phase angles. Symmetries in the problem suggest that preferred rotation states are likely, defined relative to the object-sun direction in inertial space and relative to the maximum principal axis of inertia in the body coordinate system. Such rotation states may greatly simplify the problem of light curve interpretation by reducing the number of degrees of freedom in the problem.

Ojakangas, Gregory W.↗
NASA NTRS2004

Compact Optoelectronic Compass

A compact optoelectronic sensor unit measures the apparent motion of the Sun across the sky. The data acquired by this chip are processed in an external processor to estimate the relative orientation of the axis of rotation of the Earth. Hence, the combination of this chip and the external processor finds the direction of true North relative to the chip: in other words, the combination acts as a solar compass. If the compass is further combined with a clock, then the combination can be used to establish a threeaxis inertial coordinate system. If, in addition, an auxiliary sensor measures the local vertical direction, then the resulting system can determine the geographic position. This chip and the software used in the processor are based mostly on the same design and operation as those of the unit described in Micro Sun Sensor for Spacecraft (NPO-30867) elsewhere in this issue of NASA Tech Briefs. Like the unit described in that article, this unit includes a small multiple-pinhole camera comprising a micromachined mask containing a rectangular array of microscopic pinholes mounted a short distance in front of an image detector of the active-pixel sensor (APS) type (see figure). Further as in the other unit, the digitized output of the APS in this chip is processed to compute the centroids of the pinhole Sun images on the APS. Then the direction to the Sun, relative to the compass chip, is computed from the positions of the centroids (just like a sundial). In the operation of this chip, one is interested not only in the instantaneous direction to the Sun but also in the apparent path traced out by the direction to the Sun as a result of rotation of the Earth during an observation interval (during which the Sun sensor must remain stationary with respect to the Earth). The apparent path of the Sun across the sky is projected on a sphere. The axis of rotation of the Earth lies at the center of the projected circle on the sphere surface. Hence, true North (not magnetic North), relative to the chip, can be estimated from paths of the Sun images across the APS. In a test, this solar compass has been found to yield a coarse estimate of the North (within tens of degrees) in an observation time of about ten minutes. As expected, the accuracy was found to increase with observation time: after a few hours, the estimated direction of the rotation axis becomes accurate to within a small fraction of a degree.

Christian, Carl↗
NASA NTRS2004

Reducing Errors by Use of Redundancy in Gravity Measurements

A methodology for improving gravity-gradient measurement data exploits the constraints imposed upon the components of the gravity-gradient tensor by the conditions of integrability needed for reconstruction of the gravitational potential. These constraints are derived from the basic equation for the gravitational potential and from mathematical identities that apply to the gravitational potential and its partial derivatives with respect to spatial coordinates. Consider the gravitational potential in a Cartesian coordinate system {x1,x2,x3}. If one measures all the components of the gravity-gradient tensor at all points of interest within a region of space in which one seeks to characterize the gravitational field, one obtains redundant information. One could utilize the constraints to select a minimum (that is, nonredundant) set of measurements from which the gravitational potential could be reconstructed. Alternatively, one could exploit the redundancy to reduce errors from noisy measurements. A convenient example is that of the selection of a minimum set of measurements to characterize the gravitational field at n3 points (where n is an integer) in a cube. Without the benefit of such a selection, it would be necessary to make 9n3 measurements because the gravitygradient tensor has 9 components at each point. The problem of utilizing the redundancy to reduce errors in noisy measurements is an optimization problem: Given a set of noisy values of the components of the gravity-gradient tensor at the measurement points, one seeks a set of corrected values - a set that is optimum in that it minimizes some measure of error (e.g., the sum of squares of the differences between the corrected and noisy measurement values) while taking account of the fact that the constraints must apply to the exact values. The problem as thus posed leads to a vector equation that can be solved to obtain the corrected values.

Kulikov, Igor↗
NASA NTRS2011

Vertical Distribution of Aersols and Water Vapor Using CRISM Limb Observations

Near-infrared spectra taken in a limb-viewing geometry by the Compact Reconnaissance Imaging Spectrometer for Mars (CRISM) on-board the Mars Reconnaissance Orbiter (MRO) provide a useful tool for probing atmospheric structure. Specifically, the observed radiance as a function of wavelength and height above the limb allows the vertical distribution of both dust and ice aerosols to be retrieved. These data serve as an important supplement to the aerosol profiling provided by the MRO/MCS instrument allowing independent validation and giving additional information on particle physical and scattering properties through multi-wavelength studies. A total of at least ten CRISM limb observations have been taken so far covering a full Martian year. Each set of limb observations nominally contains about four dozen scans across the limb giving pole-to-pole coverage for two orbits at roughly 100 and 290 W longitude over the Tharsis and Syrtis/Hellas regions, respectively. At each longitude, limb scans are spaced roughly 10 degrees apart in latitude, with a vertical spatial resolution on the limb of roughly 800 m. Radiative transfer modeling is used to model the observations. We compute synthetic CRISM limb spectra using a discrete-ordinates radiative transfer code that accounts for multiple scattering from aerosols and accounts for spherical geometry of the limb observations by integrating the source functions along curved paths in that coordinate system. Retrieved are 14-point vertical profiles for dust and water ice aerosols with resolution of 0.4 scale heights between one and six scale heights above the surface. After the aerosol retrieval is completed, the abundances of C02 (or surface pressure) and H20 gas are retrieved by matching the depth of absorption bands at 2000 nm for carbon dioxide and at 2600 run for water vapor. In addition to the column abundance of water vapor, limited information on its vertical structure can also be retrieved depending on the signal available from aerosol scattering.

Smith, Michael D.↗
NASA NTRS2003

yourSky: Custom Sky-Image Mosaics via the Internet

yourSky (http://yourSky.jpl.nasa.gov) is a computer program that supplies custom astronomical image mosaics of sky regions specified by requesters using client computers connected to the Internet. [yourSky is an upgraded version of the software reported in Software for Generating Mosaics of Astronomical Images (NPO-21121), NASA Tech Briefs, Vol. 25, No. 4 (April 2001), page 16a.] A requester no longer has to engage in the tedious process of determining what subset of images is needed, nor even to know how the images are indexed in image archives. Instead, in response to a requester s specification of the size and location of the sky area, (and optionally of the desired set and type of data, resolution, coordinate system, projection, and image format), yourSky automatically retrieves the component image data from archives totaling tens of terabytes stored on computer tape and disk drives at multiple sites and assembles the component images into a mosaic image by use of a high-performance parallel code. yourSky runs on the server computer where the mosaics are assembled. Because yourSky includes a Web-interface component, no special client software is needed: ordinary Web browser software is sufficient.

Jacob, Joseph↗
NASA NTRS2011

In Situ Surface Characterization

Operation of in situ space assets, such as rovers and landers, requires operators to acquire a thorough understanding of the environment surrounding the spacecraft. The following programs help with that understanding by providing higher-level information characterizing the surface, which is not immediately obvious by just looking at the XYZ terrain data. This software suite covers three primary programs: marsuvw, marsrough, and marsslope, and two secondary programs, which together use XYZ data derived from in situ stereo imagery to characterize the surface by determining surface normal, surface roughness, and various aspects of local slope, respectively. These programs all use the Planetary Image Geometry (PIG) library to read mission-specific data files. The programs themselves are completely multimission; all mission dependencies are handled by PIG. The input data consists of images containing XYZ locations as derived by, e.g., marsxyz. The marsuvw program determines surface normals from XYZ data by gathering XYZ points from an area around each pixel and fitting a plane to those points. Outliers are rejected, and various consistency checks are applied. The result shows the orientation of the local surface at each point as a unit vector. The program can be run in two modes: standard, which is typically used for in situ arm work, and slope, which is typically used for rover mobility. The difference is primarily due to optimizations necessary for the larger patch sizes in the slope case. The marsrough program determines surface roughness in a small area around each pixel, which is defined as the maximum peak-to-peak deviation from the plane perpendicular to the surface normal at that pixel. The marsslope program takes a surface normal file as input and derives one of several slope-like outputs from it. The outputs include slope, slope rover direction (a measure of slope radially away from the rover), slope heading, slope magnitude, northerly tilt, and solar energy (compares the slope with the Sun s location at local noon). The marsuvwproj program projects a surface normal onto an arbitrary plane in space, resulting in a normalized 3D vector, which is constrained to lie in the plane. The marsuvwrot program rotates the vectors in a surface normal file, generating a new surface normal file. It also can change coordinate systems for an existing surface normal file. While the algorithms behind this suite are not particularly unique, what makes the programs useful is their integration into the larger in situ image processing system via the PIG library. They work directly with space in situ data, understanding the appropriate image metadata fields and updating them properly. The secondary programs (marsuvwproj, marsuvwrot) were originally developed to deal with anomalous situations on Opportunity and Spirit, respectively, but may have more general applicability.

Deen, Robert G.↗
NASA NTRS2012

Lunar Topography: Results from the Lunar Orbiter Laser Altimeter

The Lunar Orbiter Laser Altimeter (LOLA) onboard the Lunar Reconnaissance Orbiter (LRO) has been operating nearly continuously since July 2009, accumulating over 6 billion measurements from more than 2 billion in-orbit laser shots. LRO's near-polar orbit results in very high data density in the immediate vicinity of the lunar poles, with full coverage at the equator from more than 12000 orbital tracks averaging less than 1 km in spacing at the equator. LRO has obtained a global geodetic model of the lunar topography with 50-meter horizontal and 1-m radial accuracy in a lunar center-of-mass coordinate system, with profiles of topography at 20-m horizontal resolution, and 0.1-m vertical precision. LOLA also provides measurements of reflectivity and surface roughness down to its 5-m laser spot size. With these data LOLA has measured the shape of all lunar craters 20 km and larger. In the proposed extended mission commencing late in 2012, LOLA will concentrate observations in the Southern Hemisphere, improving the density of the polar coverage to nearly 10-m pixel resolution and accuracy to better than 20 m total position error. Uses for these data include mission planning and targeting, illumination studies, geodetic control of images, as well as lunar geology and geophysics. Further improvements in geodetic accuracy are anticipated from the use of re ned gravity fields after the successful completion of the Gravity Recovery and Interior Laboratory (GRAIL) mission in 2012.

Neumann, Gregory↗
NASA NTRS2012

SAR Polarimetry

Radar Scattering includes: Surface Characteristics, Geometric Properties, Dielectric Properties, Rough Surface Scattering, Geometrical Optics and Small Perturbation Method Solutions, Integral Equation Method, Magellan Image of Pancake Domes on Venus, Dickinson Impact Crater on Venus (Magellan), Lakes on Titan (Cassini Radar, Longitudinal Dunes on Titan (Cassini Radar), Rough Surface Scattering: Effect of Dielectric Constant, Vegetation Scattering, Effect of Soil Moisture. Polarimetric Radar includes: Principles of Polarimetry: Field Descriptions, Wave Polarizations: Geometrical Representations, Definition of Ellipse Orientation Angles, Scatter as Polarization Transformer, Scattering Matrix, Coordinate Systems, Scattering Matrix, Covariance Matrix, Pauli Basis and Coherency Matrix, Polarization Synthesis, Polarimeter Implementation.

radar scattering↗
NASA NTRS2012

Efficient Implementation of the Invariant Imbedding T-Matrix Method and the Separation of Variables Method Applied to Large Nonspherical Inhomogeneous Particles

Three terms, ''Waterman's T-matrix method'', ''extended boundary condition method (EBCM)'', and ''null field method'', have been interchangeable in the literature to indicate a method based on surface integral equations to calculate the T-matrix. Unlike the previous method, the invariant imbedding method (IIM) calculates the T-matrix by the use of a volume integral equation. In addition, the standard separation of variables method (SOV) can be applied to compute the T-matrix of a sphere centered at the origin of the coordinate system and having a maximal radius such that the sphere remains inscribed within a nonspherical particle. This study explores the feasibility of a numerical combination of the IIM and the SOV, hereafter referred to as the IIMþSOV method, for computing the single-scattering properties of nonspherical dielectric particles, which are, in general, inhomogeneous. The IIMþSOV method is shown to be capable of solving light-scattering problems for large nonspherical particles where the standard EBCM fails to converge. The IIMþSOV method is flexible and applicable to inhomogeneous particles and aggregated nonspherical particles (overlapped circumscribed spheres) representing a challenge to the standard superposition T-matrix method. The IIMþSOV computational program, developed in this study, is validated against EBCM simulated spheroid and cylinder cases with excellent numerical agreement (up to four decimal places). In addition, solutions for cylinders with large aspect ratios, inhomogeneous particles, and two-particle systems are compared with results from discrete dipole approximation (DDA) computations, and comparisons with the improved geometric-optics method (IGOM) are found to be quite encouraging.

Invariant imbedding method↗
NASA NTRS2015

GNSS Ephemeris with Graceful Degradation and Measurement Fusion

A method for providing an extended propagation ephemeris model for a satellite in Earth orbit, the method includes obtaining a satellite's orbital position over a first period of time, applying a least square estimation filter to determine coefficients defining osculating Keplarian orbital elements and harmonic perturbation parameters associated with a coordinate system defining an extended propagation ephemeris model that can be used to estimate the satellite's position during the first period, wherein the osculating Keplarian orbital elements include semi-major axis of the satellite (a), eccentricity of the satellite (e), inclination of the satellite (i), right ascension of ascending node of the satellite (.OMEGA.), true anomaly (.theta.*), and argument of periapsis (.omega.), applying the least square estimation filter to determine a dominant frequency of the true anomaly, and applying a Fourier transform to determine dominant frequencies of the harmonic perturbation parameters.

Garrison, James Levi↗
NASA NTRS1961

Coordinates for Mapping the Distribution of Magnetically Trapped Particles

Dipole representations of the earth's magnetic field have been found to have insufficient accuracy for the study of magnetically trapped particles. A coordinate system consisting of the magnitude of the magnetic field B, and the integral invariant I has been found to adequately organize, measurements made at different geographic locations. It is shown in the present paper, that a parameter L = f(B,I) can be defined which retains most of the desirable properties of I and which has the additional property of organizing measurements along lines of force. Since the parameter L is the analog of a physical distance in a dipole field (the equatorial radius of a magnetic shell), it is usually found to present fewer conceptual difficulties than the integral invariant I.

McIlwain, Carl E.↗
NASA NTRS2016

Trajectory Design Employing Convex Optimization for Landing on Irregularly Shaped Asteroids

Mission proposals that land on asteroids are becoming popular. However, in order to have a successful mission the spacecraft must reliably and softly land at the intended landing site. The problem under investigation is how to design a fuel-optimal powered descent trajectory that can be quickly computed on- board the spacecraft, without interaction from ground control. An optimal trajectory designed immediately prior to the descent burn has many advantages. These advantages include the ability to use the actual vehicle starting state as the initial condition in the trajectory design and the ease of updating the landing target site if the original landing site is no longer viable. For long trajectories, the trajectory can be updated periodically by a redesign of the optimal trajectory based on current vehicle conditions to improve the guidance performance. One of the key drivers for being completely autonomous is the infrequent and delayed communication between ground control and the vehicle. Challenges that arise from designing an asteroid powered descent trajectory include complicated nonlinear gravity fields, small rotating bodies and low thrust vehicles. There are two previous studies that form the background to the current investigation. The first set looked in-depth at applying convex optimization to a powered descent trajectory on Mars with promising results.1, 2 This showed that the powered descent equations of motion can be relaxed and formed into a convex optimization problem and that the optimal solution of the relaxed problem is indeed a feasible solution to the original problem. This analysis used a constant gravity field. The second area applied a successive solution process to formulate a second order cone program that designs rendezvous and proximity operations trajectories.3, 4 These trajectories included a Newtonian gravity model. The equivalence of the solutions between the relaxed and the original problem is theoretically established. The proposed solution for designing the asteroid powered descent trajectory is to use convex optimization, a gravity model with higher fidelity than Newtonian, and an iterative solution process to design the fuel optimal trajectory. The solution to the convex optimization problem is the thrust profile, magnitude and direction, that will yield the minimum fuel trajectory for a soft landing at the target site, subject to various mission and operational constraints. The equations of motion are formulated in a rotating coordinate system and includes a high fidelity gravity model. The vehicle's thrust magnitude can vary between maximum and minimum bounds during the burn. Also, constraints are included to ensure that the vehicle does not run out of propellant, or go below the asteroid's surface, and any vehicle pointing requirements. The equations of motion are discretized and propagated with the trapezoidal rule in order to produce equality constraints for the optimization problem. These equality constraints allow the optimization algorithm to solve the entire problem, without including a propagator inside the optimization algorithm.

Pinson, Robin M.↗